Topics in the arithmetic of del Pezzo and K3 surfaces

Battleship is one of the most common games among kids. It consists of a small plane grid, endowed with coordinates, and two players that have to guess the position of each other's ships placed on the grid. To describe my research in non-mathematical terms, I would say that what I do is considering an infinite 3D battleship grid; while in the game a ship is a bunch of squares put on the grid, in my case a ship is a whole surface; also, in the game two coordinates denote a square of the grid, here three coordinates denote a point of the grid. Then, given a surface in a 3D grid, one might ask which points of the grid lie on the surface, how many of them, or whether there is any.

The problems I considered only deal with surfaces satisfying some particular conditions: for example, they must have no holes. The surfaces involved in my research fall into two big classes of surfaces, K3 surfaces and del Pezzo surfaces. In my research I filled in some gaps in the work or replied to the questions of other mathematicians about the rational points of the surface considered, i.e., the grid's points lying on the surface.

Abstract

In this thesis we study the unirationality of del Pezzo surfaces of degree 2 over finite fields, proving that every such surface is unirational. We explicitly compute the Picard lattice of the members of a 1-dimensional family of K3 surfaces. We produce an explicit example of a K3 surface having a particular Picard lattice of rank 2.